A057972 Number of 5 X n binary matrices with 3 unit columns up to row and column permutations.

3, 31, 252, 1776, 11048, 61106, 303664, 1368844, 5651241, 21559133, 76613440, 255411923, 803771681, 2400633464, 6837010458, 18644075466, 48855805143, 123415815229, 301386128354, 713271875603, 1639572164669, 3667859207856

Offset: 3

Author

Vladeta Jovovic, Oct 21 2000

Keywords

• nonn

Comments

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 8 points of that set uniquely (if offset is 8).

Links

• V. Jovovic, Number of minimal covers of an unlabeled n - set that cover k points of that set uniquely
• V. Jovovic, Number of binary matrices with fixed number of unit columns up to row and column permutations

Crossrefs

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057969 - A057971.

Formula

Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) + 20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

G.f. : x^3/120*(35/(1 - x^1)^27 + 130/(1 - x^1)^13/(1 - x^2)^7 + 45/(1 - x^1)^7/(1 - x^2)^10 + 100/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).